Minimax Problem Research Articles

Designing incoherent unit norm tight frames via block coordinate descent-based alternating projection

In this paper, we introduce a novel methodology for constructing incoherent unit norm tight frames, termed block coordinate descent-based alternating projection. In the proposed approach, the underlying min-max problem was reformulated in a block-wise fashion by taking a fixed number of columns as a block and adding a spectral constraint, and it was approximately solved via an alternating projection method with Nesterov accelerated gradient technique. Numerical experiments demonstrate that the resulting algorithm is particularly efficient for the construction of incoherent unit norm tight frames with high redundancy and has great advantages in running efficiency.

Jointly Improving the Sample and Communication Complexities in Decentralized Stochastic Minimax Optimization

We propose a novel single-loop decentralized algorithm, DGDA-VR, for solving the stochastic nonconvex strongly-concave minimax problems over a connected network of agents, which are equipped with stochastic first-order oracles to estimate their local gradients. DGDA-VR, incorporating variance reduction, achieves O(ε^−3) oracle complexity and O(ε^−2) communication complexity without resorting to multi-communication rounds – both are optimal, i.e., matching the lower bounds for this class of problems. Since DGDA-VR does not require multiple communication rounds, it is applicable to a broader range of decentralized computational environments. To the best of our knowledge, this is the first distributed method using a single communication round in each iteration to jointly optimize the oracle and communication complexities for the problem considered here.

Solving Maxmin Optimization Problems via Population Games

Population games are games with a finite set of available strategies and an infinite number of players, in which the reward for choosing a given strategy is a function of the distribution of players over strategies. The paper shows that, in a certain class of maxmin optimization problems, it is possible to associate a population game to a given maxmin problem in such a way that solutions to the optimization problem are found from Nash equilibria of the associated game. Iterative solution methods for maxmin optimization problems can then be derived from systems of differential equations whose trajectories are known to converge to Nash equilibria. In particular, we use a discrete-time version of the celebrated replicator equation of evolutionary game theory, also known in machine learning as the exponential multiplicative weights algorithm. The resulting algorithm can be viewed as a generalization of the Iteratively Reweighted Least Squares (IRLS) method, which is well known in numerical analysis as a useful technique for solving Chebyshev function approximation problems on a finite grid. Examples are provided to show the use of the generalized IRLS method in collective investment and in decision making under model uncertainty.

Statistical inference for generative adversarial networks and other minimax problems

AbstractThis paper studies generative adversarial networks (GANs) from the perspective of statistical inference. A GAN is a popular machine learning method in which the parameters of two neural networks, a generator and a discriminator, are estimated to solve a particular minimax problem. This minimax problem typically has a multitude of solutions and the focus of this paper are the statistical properties of these solutions. We address two key statistical issues for the generator and discriminator network parameters, consistent estimation and confidence sets. We first show that the set of solutions to the sample GAN problem is a (Hausdorff) consistent estimator of the set of solutions to the corresponding population GAN problem. We then devise a computationally intensive procedure to form confidence sets and show that these sets contain the population GAN solutions with the desired coverage probability. Small numerical experiments and a Monte Carlo study illustrate our results and verify our theoretical findings. We also show that our results apply in general minimax problems that may be nonconvex, nonconcave, and have multiple solutions.

Sum-of-squares relaxations for polynomial min–max problems over simple sets

Sum-of-squares relaxations for polynomial min–max problems over simple sets

Blahut-Arimoto Algorithms for Inner and Outer Bounds on Capacity Regions of Broadcast Channels.

The celebrated Blahut-Arimoto algorithm computes the capacity of a discrete memoryless point-to-point channel by alternately maximizing the objective function of a maximization problem. This algorithm has been applied to degraded broadcast channels, in which the supporting hyperplanes of the capacity region are again cast as maximization problems. In this work, we consider general broadcast channels and extend this algorithm to compute inner and outer bounds on the capacity regions. Our main contributions are as follows: first, we show that the optimization problems are max-min problems and that the exchange of minimum and maximum holds; second, we design Blahut-Arimoto algorithms for the maximization part and gradient descent algorithms for the minimization part; third, we provide convergence analysis for both parts. Numerical experiments validate the effectiveness of our algorithms.

Alternating and Parallel Proximal Gradient Methods for Nonsmooth, Nonconvex Minimax: A Unified Convergence Analysis

There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting. Funding: The research of E. Cohen was partially supported by a doctoral fellowship from the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240]. The research of M. Teboulle was partially supported by the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240].

SCSO-MHEF: Sand Cat Swarm Optimization based MHEF for Nonlinear LTI-IoT Sensor Data Enhancement

Sensor data is an integral component of internet of things (IoT) and edge computing environments and initiatives. In IoT, almost any entity imaginable can be outfitted with a unique identifier and the capacity to transfer data over a network. The estimate problem was formulated as a min-max problem subject to system dynamics and limitations on states and disturbances within the moving horizon strategy framework. In this paper, a novel Sand Cat Swarm Optimization Based MHEF for Nonlinear LTI IOT Sensor Data Enhancement (SCSO-MHEF) is proposed. In the proposed method the MHEF is optimized using Sand Cat Swarm Optimization to enhance sensor data stability tuned by initial parameters. Simulation experiments were conducted on various and unique scenarios in various orders LTI system with IOT sensor data in order to validate the suggested approach. This method can be used to analyze systems with dynamically changing systems. The proposed SCSO-MHEF technique overall accuracy of 84.5%, 87.3 %, and 99.5 % better than Kalman Filter (KF), EKF and Moving Horizon Filter (MHEF) respectively.

SCSO-MHEF: Sand Cat Swarm Optimization based MHEF for Nonlinear LTI-IoT Sensor Data Enhancement

Sensor data is an integral component of internet of things (IoT) and edge computing environments and initiatives. In IoT, almost any entity imaginable can be outfitted with a unique identifier and the capacity to transfer data over a network. The estimate problem was formulated as a min-max problem subject to system dynamics and limitations on states and disturbances within the moving horizon strategy framework. In this paper, a novel Sand Cat Swarm Optimization Based MHEF for Nonlinear LTI IOT Sensor Data Enhancement (SCSO-MHEF) is proposed. In the proposed method the MHEF is optimized using Sand Cat Swarm Optimization to enhance sensor data stability tuned by initial parameters. Simulation experiments were conducted on various and unique scenarios in various orders LTI system with IOT sensor data in order to validate the suggested approach. This method can be used to analyze systems with dynamically changing systems. The proposed SCSO-MHEF technique overall accuracy of 84.5%, 87.3 %, and 99.5 % better than Kalman Filter (KF), EKF and Moving Horizon Filter (MHEF) respectively.

Adversarial Decision Making Against Intelligent Targets in Cooperative Multiagent Systems

In this paper we propose a distributed adversarial decision making approach for multi-agent system (MAS), which is supposed to detect a team of intelligent targets. The MAS is demanded to achieve the best detection benefit against the intelligent targets that can shelter part of their features and prevent the detection according to an expert defense policy. One of the key challenges is how to achieve higher detection benefit that is both determined by the target assignment action of the MAS and the non-predictable defense policy of the targets. To handle this problem, we first formulate the multi-agent decision making problem as a max-min problem of detection benefit and break it into separate parts that are easier to be optimized. Then we introduce a new variant of distributed alternating direction method of multipliers (ADMM) to search the optimal solutions under the worst defense policy that the targets choose. To overcome the lack of access to global convergence of multi-block ADMM, we add local additional variables to formulate a penalty for non-convex parts of the local objective function. The convergence to an equilibrium and the optimality of the detection benefit are empirically validated by numerical simulations. The influence of the parameter setting is also presented and can be regarded as a prior suggestion for real applications.

RS-DRL-based offloading policy and UAV trajectory design in F-MEC systems

For better flexibility and greater coverage areas, Unmanned Aerial Vehicles (UAVs) have been applied in Flying Mobile Edge Computing (F-MEC) systems to offer offloading services for the User Equipment (UEs). This paper considers a disaster-affected scenario where UAVs undertake the role of MEC servers to provide computing resources for Disaster Relief Devices (DRDs). Considering the fairness of DRDs, a max-min problem is formulated to optimize the saved time by jointly designing the trajectory of the UAVs, the offloading policy and serving time under the constraint of the UAVs' energy capacity. To solve the above non-convex problem, we first model the service process as a Markov Decision Process (MDP) with the Reward Shaping (RS) technique, and then propose a Deep Reinforcement Learning (DRL) based algorithm to find the optimal solution for the MDP. Simulations show that the proposed RS-DRL algorithm is valid and effective, and has better performance than the baseline algorithms.

Some Optimality Conditions for Minimax Problem

Some Optimality Conditions for Minimax Problem

Novel Continuous- and Discrete-Time Neural Networks for Solving Quadratic Minimax Problems With Linear Equality Constraints.

This article presents two novel continuous- and discrete-time neural networks (NNs) for solving quadratic minimax problems with linear equality constraints. These two NNs are established based on the conditions of the saddle point of the underlying function. For the two NNs, a proper Lyapunov function is constructed so that they are stable in the sense of Lyapunov, and will converge to some saddle point(s) for any starting point under some mild conditions. Compared with the existing NNs for solving quadratic minimax problems, the proposed NNs require weaker stability conditions. The validity and transient behavior of the proposed models are illustrated by some simulation results.

Robust Radar Waveform Optimization Under Target Interpulse Fluctuation and Practical Constraints via Sequential Lagrange Dual Approximation

This correspondence considers the problem of robust radar waveform design against target interpulse fluctuation with constraints on the transmit energy, similarity, and signal dynamic range. We aim to make robust the transmit waveform and receive filter with respect to the Target Amplitude Vector (TAV) uncertainties by enhancing the worst-case Signal-to-Noise-Ratio (SNR) over the modeled ellipsoidal set. First of all, the filter is directly solved leveraging on the existence of saddle point condition, which leads to a suitable simplification of the original non-convex problem. Then, to handle the resultant max-min problem, the Sequential Lagrange Dual Approximation (SLDA) procedure is devised. Specifically, we update the transmit waveform via solving a sequence of max-min approximate problems, which are proven to be hidden convex relying on the theory of Lagrange duality. Finally, the filter is synthesized after solving a Quadratical Constraint Quadratic Programming (QCQP) problem based on the optimized waveform. Numerical results highlight the robustness of the proposed design which guarantees an enhanced worst-case SNR with respect to TAV uncertainties.

Entangling capacity of operators

Given a unitary operator U acting on a composite quantum system what is the entangling capacity of U? This question is investigated using a geometric approach. The entangling capacity, defined via metrics on the unitary groups, leads to a minimax problem. The dual, a maximin problem, is investigated in parallel and yields some familiar entanglement measures. A class of entangling operators, called generalized control operators is defined. The entangling capacities and other properties for this class of operators is studied.

252 Imprecision in Nutrition Leads to More Desired Feed Intake in Growing and Finishing Pigs

Abstract There is a general consensus that analyzing nutrients in feed as best as possible will lead to better animal performance and reduced costs for the farmer. Precision is defined by how close repeated nutrient measurements in the same feed agree with each other. From an economical point of view, improving the precision of all nutrients in the feed as best as possible is sub optimal, as analyzing nutrients in feed takes time and increases costs. We show that imprecision of nutrients in feed can be linked to over-eating in growing and finishing pigs, hence reduced profit for the farmer. Therefore, we solved a min-max problem, where we minimize the cost and time analyzing nutrients in feed, while maximizing the profit for the farmer to find the best economical precision. We propose a mathematical model and use the NRC Swine requirement system model [NRC (2012)] to connect the daily feed intake of the animal with the precision of the nutrients in the feed. The model assumes that the pig will eat ad libitum for maximal potential and will not be hindered by physical or environmental restrictions. It is assumed that the animal will eat an amount of feed sufficient to attain its potential growth [Emmans (1981)]. The (desired) feed intake is calculated from the requirements of the pig and estimated energy and amino acids in the feed. The true feed intake will be different from the estimated feed intake because there are precision errors in the estimation of the energy and amino acids in the feed. As we cannot know the true nutrient values and true feed intake, we must consider the estimated nutrient values and feed intake as stochastic variables. Using Monte-Carlo we simulate the stochastic feed intake and calculate the average difference between the expected and true feed intake. The Monte-Carlo simulations demonstrated two interesting results. Firstly, growing and finishing swine will eat, on average, more than predicted, when precision is above 1.5% (P < 0.03). Secondly, an increase in precision in nutrient composition of diets yields a decreased excessive feed intake, but shows diminishing returns when precision becomes small. We estimate that economically the best precision of nutrient estimations in the feed is below 2%. In the case of 2% precision, pigs will, on average, eat between 0.065 kg and 0.497 kg more than expected (P = 0.05), while the compound feed producer will not, on average, add more nutrition in the feed than needed (P = 0.01).

Two-stage robust optimal capacity configuration of a wind, photovoltaic, hydropower, and pumped storage hybrid energy system

The hybrid energy system of hydro-powers, pumped storages and renewable energies has become a new topic direction in modern power system developments. Consequently, it is essential to realize a rational and efficient allocation of different energy source capacities. Nevertheless, there is still a gap between the available studies and the requirement for further hybrid energy system development. This paper focuses on the optimal capacity configuration of a wind, photovoltaic, hydropower, and pumped storage power system. In this direction, a bi-level programming model for the optimal capacity configuration of wind, photovoltaic, hydropower, pumped storage power system is derived. To model the operating mode of a pumped storage power station, two 0-1 variables are introduced. To handle the nonlinear and nonconvex lower level programing problem caused by the two 0-1 variables, it is proposed that the 0-1 variables are treated as some uncertain parameters. Also, by treating the 0-1 variables as some uncertain parameters, a two-stage robust optimization problem to decompose the original bi-level programing one into a master problem and a subproblem is finally introduced. The Karush-Kuhn-Tucker (KKT) conditions are then applied to simplify and linearize the min-max problem and nonlinear terms in the master problem. This results in both the master problem and the subproblem being formulated as mixed integer linear programming (MILP) problems. By utilizing the powerful Column-and-Constraint Generation (C&CG) algorithm, the two-stage robust optimization model is decomposed into an iterative procedure of solving the master problem and the subproblem sequentially. This approach eliminates the need for intricate optimization algorithms as commonly used in existing bi-level planning problems in hybrid energy systems. Finally, the effectiveness and advantages of the proposed model is verified by the numerical results on a case study.

Minimax Problems with Coupled Linear Constraints: Computational Complexity and Duality

Minimax Problems with Coupled Linear Constraints: Computational Complexity and Duality

Network-constrained flexible ramping product provision of prosumer aggregator: a data-driven stochastic bi-level optimization

Prosumers are expected to provide the flexible ramping product (FRP) in the power system. However, voltage violations and line congestion may arise in the distribution network, when FRP delivered by prosumers. Hence, this paper proposes a data-driven stochastic bi-level optimization model to coordinate the prosumer aggregator to decide FRP-offering while ensuring distribution network security under FRP delivery. In the proposed bi-level model, the upper-level is a min-max problem, representing the minimum expected cost under the worst-case scenario probability distribution for the prosumer aggregator. The lower-level is the operation cost minimization within the distribution network security for distribution network operator. The proposed model is converted into a single-level model using the Karush-Kuhn-Tucker condition and strong duality theory, and applied to the modified IEEE 33-bus network with three prosumers. The results demonstrate the effectiveness of the proposed model.

Newton and interior-point methods for (constrained) nonconvex–nonconcave minmax optimization with stability and instability guarantees

We address the problem of finding a local solution to a nonconvex–nonconcave minmax optimization using Newton type methods, including primal-dual interior-point ones. The first step in our approach is to analyze the local convergence properties of Newton’s method in nonconvex minimization. It is well established that Newton’s method iterations are attracted to any point with a zero gradient, irrespective of it being a local minimum. From a dynamical system standpoint, this occurs because every point for which the gradient is zero is a locally asymptotically stable equilibrium point. We show that by adding a multiple of the identity such that the Hessian matrix is always positive definite, we can ensure that every non-local-minimum equilibrium point becomes unstable (meaning that the iterations are no longer attracted to such points), while local minima remain locally asymptotically stable. Building on this foundation, we develop Newton-type algorithms for minmax optimization, conceptualized as a sequence of local quadratic approximations for the minmax problem. Using a local quadratic approximation serves as a surrogate for guiding the modified Newton’s method toward a solution. For these local quadratic approximations to be well-defined, it is necessary to modify the Hessian matrix by adding a diagonal matrix. We demonstrate that, for an appropriate choice of this diagonal matrix, we can guarantee the instability of every non-local-minmax equilibrium point while maintaining stability for local minmax points. Using numerical examples, we illustrate the importance of guaranteeing the instability property. While our results are about local convergence, the numerical examples also indicate that our algorithm enjoys good global convergence properties.